Mod p Principal Series Representations

• Mod p principal series representations are defined via compact induction from a Borel subgroup over fields of characteristic p and exhibit distinct structural and homological properties.
• They are analyzed using advanced tools like Bruhat filtrations, derived functors, and spectral sequences to classify irreducibility and submodule structures.
• These representations are pivotal to the mod p Langlands program, linking group theoretical insights with arithmetic cohomology and functorial constructions.

The theory of mod $p$ principal series representations occupies a central position in the modular representation theory of reductive groups and their covers over local and finite fields. These representations, defined over coefficient fields of characteristic $p$, exhibit unique structural, homological, and categorical properties that distinguish them from their characteristic zero analogues and have deep connections with questions in the mod $p$ local Langlands program, cohomology of arithmetic groups, and $p$-adic Hodge theory.

1. Definitions and Construction

Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ a connected split reductive group over $F$. Fix a Borel subgroup $B \subset G$ with unipotent radical $N$ and Levi torus $T=B\cap B^-$. For a coefficient field $k_E$ of characteristic $p$, and a smooth character $\chi:T(F)\rightarrow k_E^\times$, the smooth mod $p$ principal series $\operatorname{Ind}_{B^-(F)}^{G(F)} \chi$ is the space of locally constant functions $f:G(F)\to k_E$ satisfying $f(b^-g)=\chi(b^-)\,f(g)$ for all $b^-\in B^-(F)$, $g\in G(F)$, with compact support modulo $B^-(F)$. This is a smooth admissible $G(F)$-module over $k_E$ (Hauseux, 2013, Koziol, 2017). For groups over finite fields, e.g., $G_q=\mathrm{GL}_2(\mathbb{F}_q)$, analogous constructions yield $$\operatorname{PS}(\chi_1,\chi_2):=\operatorname{Ind}_{B_q}^{G_q}(\chi_1\otimes\chi_2)$$ (Ghate et al., 17 Jun 2025).

The principal series concept generalizes to various contexts:

• Principal series for finite rings $G_n=\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^n)$ (Schein et al., 6 Nov 2025)
• Genuine principal series for metaplectic covers $\widetilde{G}$ of $G$ (Koziol et al., 2016, Witthaus, 2022, Peskin, 2014)

In the metaplectic or covering group context (split, type $C_n$, $p\neq 2$), a smooth character $\sigma:T\to\Bbbk^\times$ is "genuine" after twisting by a certain character $\chi_\psi$ arising from the Weil index and additive character of $F$, and the principal series is defined via induction from the lifted torus character to the cover (Koziol et al., 2016).

2. Filtrations, Bruhat Theory, and Homological Tools

Mod $p$ principal series are analyzed using several filtrations and functorial constructions:

• Bruhat Filtration: The classical Bruhat stratification of $\operatorname{Ind}\,U$ as a $B(F)$-representation enables precise control of submodule structure and explicit calculation of Ext-groups. If $d = \dim N$, the filtration $$0=I_{-1}\subset I_0\subset \dots\subset I_d = \operatorname{Ind}\,U$$ has graded pieces $$I_r/I_{r-1} \cong \bigoplus_{\ell(w)=r} C_c(N_w(F), U)$$, where $N_w=N\cap w^{-1}N w$ (Hauseux, 2013).
• Ordinary Parts and $\delta$-Functor Techniques: Emerton's derived ordinary parts functor $H^\bullet \mathrm{Ord}_{B(F)}$ provides a delta-functor from admissible $G(F)$-representations to those of $T(F)$, yielding spectral sequences that express higher Ext-groups between principal series in terms of the torus and its derived functors. This equips the category of mod $p$ representations of $G(F)$ with powerful homological control (Hauseux, 2013).
• Socle and Radical Filtrations: For principal series over finite rings, explicit "types" (e.g., for $G_2=\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^2)$) and carry-set theory parameterize submodule lattices, and adjacency in the poset of types determines the socle and radical gradations (Schein et al., 6 Nov 2025).

These tools enable the computation of projective resolutions (e.g., via the Schneider–Stuhler coefficient systems) and the deduction of vanishing results for higher Ext and cohomology (Ollivier, 2014).

3. Irreducibility Criteria and Submodule Structure

The irreducibility properties of mod $p$ principal series differ sharply from characteristic zero:

• Genericity Conditions: A character $\chi$ of $T(F)$ is weakly generic if for every simple root $\alpha$, $s_\alpha(\chi)\neq\chi$; strongly generic if $w(\chi)\neq\chi$ for all nontrivial $w\in W$ (Hauseux, 2013). Failure of (strong) genericity leads to additional (often self-) extensions and to "accidental" non-split submodules.
• Explicit Classification Over Finite Rings: In $\mathrm{GL}_2(\mathcal{O}/\mathfrak{m}^2)$, the submodule lattice and Jordan–Hölder factors are determined combinatorially by types, with linear Hasse diagrams in the totally-ramified case and infinite lattices in the non-totally-ramified case (Schein et al., 6 Nov 2025).
• Metaplectic Covers: In $\widetilde{\mathrm{Sp}}_{2n}(F)$, the length of a genuine mod $p$ principal series is $2^{m}$, where $m$ is the number of short simple roots killed by the character, and is irreducible precisely when the underlying character is nontrivial on each short coroot-lift (Koziol et al., 2016). This extends the irreducibility in the $\widetilde{\mathrm{SL}}_2(F)$ case, where all (genuine) principal series are irreducible (Peskin, 2014).

4. Extension Groups and Homological Results

Extensions between mod $p$ principal series representations are controlled by Weyl group combinatorics and derived functor calculations:

• Yoneda Ext-groups: For $G$ split with simple roots $\Delta$ and characters $\chi,\chi'$, one has $$\operatorname{Ext}^1_{G(F)}(\operatorname{Ind}\chi',\operatorname{Ind}\chi)\neq0$$ only if $\chi'=\chi$ or $\chi'=s_\alpha(\chi)$ for some $\alpha\in\Delta$, and the corresponding extension space is one-dimensional for generic $\chi$ (Hauseux, 2013).
• Spectral Sequence Realization: The spectral sequence from the (Ind,Ord) adjunction allows identification of extensions in terms of ordinary parts of the induced representation. In degree one,

$$0 \to \operatorname{Ext}^1_{T(F)}(U, \operatorname{Ord} V) \to \operatorname{Ext}^1_{G(F)}(\operatorname{Ind}U, V) \to \operatorname{Hom}_{T(F)}(U, H^1\operatorname{Ord}_{B(F)}(V))$$

and the ordinary parts can be computed explicitly using the Bruhat filtration (Hauseux, 2013).

• Connection with Mod $p$ Langlands: These extension classes are expected to coincide with those predicted by (modular) local Langlands correspondences for generic principal series parameters, and the unique non-split extensions given by simple reflections correspond to extensions within blocks associated to the same parameter (Hauseux, 2013). For $\mathrm{GL}_n$, higher cohomology of pro-$p$ Iwahori invariants reveals supersingular summands of categorical significance for local Langlands conjectures (Koziol, 2017).

5. Functors and Cohomological Constructions

The structure of mod $p$ principal series is intricately tied to various functorial constructions:

• Schneider–Vignéras Functor: This functor associates to a $B$-representation a module over the Iwasawa algebra $\Lambda(N_0)$ (with $N_0$ a compact open subgroup of the unipotent radical). For irreducible principal series, the Schneider–Vignéras module is controlled by the top Bruhat stratum, and only this stratum supports an étale $(\phi,T)$-module structure relevant for $p$-adic Galois representations (Erdélyi, 2014).
• Resolutions via Bruhat–Tits Buildings: Over arbitrary characteristic, the principal series admits a finite-length explicit projective resolution by coefficient systems on the semisimple building, a property not shared by (most) supercuspidal mod $p$ representations (Ollivier, 2014). This is crucial for computation of derived functors and for the realization of principal series as $H_0$ of the building complex.
• Cohomology of Pro-$p$-Iwahori Subgroups: The cohomology $H^1(I_1, \pi)$, for $\pi$ principal series and $I_1$ the pro-$p$-Iwahori, exhibits a filtration whose graded pieces reflect the modular representation theory of Levi subgroups and reveal occurrences of supersingular modules (Koziol, 2017).

6. Restriction, Branching Laws, and Finiteness

Branching rules for restriction of mod $p$ principal series exhibit rich combinatorial patterns:

• Finite Groups: The restriction of $\operatorname{Ind}_{B_q}^{G_q}\chi_r$ from $G_q=\mathrm{GL}_2(\mathbb{F}_q)$ to $G_p=\mathrm{GL}_2(\mathbb{F}_p)$ decomposes into a direct sum of principal series, toral, and Steinberg-twisted summands, with multiplicities dependent on the parity of the degree $f$ of the extension and explicit formulas given via Mackey theory and analysis of orbits in projective space (Ghate et al., 17 Jun 2025).
• Multiplicity and Irreducibility: The unique principal series summand occurs with multiplicity one; other summands (Steinberg-twisted, toral) have explicit multiplicities, and outside $\mathbb{F}_p$ the restriction is never irreducible (Ghate et al., 17 Jun 2025).
• Metaplectic and Covering Groups: Classification of irreducible admissible genuine mod $p$ representations for covers such as $\widetilde{\mathrm{Sp}}_{2n}(F)$ or $\widetilde{\mathrm{GL}}_2(\mathbb{Q}_p)$ proceeds via categorical equivalences between genuine modules and modules over appropriate Hecke algebras, with principal series and supersingular objects corresponding to distinct blocks (Koziol et al., 2016, Witthaus, 2022, Peskin, 2014).

7. Connections to Mod $p$ Langlands Correspondence and Socle Structure

Mod $p$ principal series are essential components in the emerging mod $p$ local Langlands correspondences and in the understanding of socle and extension structures of completed cohomology and Galois representations:

• Blocks and Socle Gradations: For $G=\mathrm{GL}_n$ and generic principal series, extension classes and socle gradations realized in completed cohomology match the predictions of topological and diagrammatic models of mod $p$ Langlands (Hauseux, 2013, Schein et al., 6 Nov 2025).
• Supersingular Constituents: For $n\geq3$, cohomological calculations show that supersingular constituents are unavoidable in the $H^1$ of pro-$p$-Iwahori, supporting the expectation that principal series blocks "see" the entire range of modular phenomena necessary for derived mod $p$ local Langlands (Koziol, 2017).
• Functorial Lifts and Galois Parameters: The Schneider–Vignéras module of a principal series provides the input for constructing étale $(\phi,\Gamma)$-modules, thus situating mod $p$ principal series at the interface of representation theory and arithmetic geometry (Erdélyi, 2014).

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This synthesis covers the construction, structural theory, homological, and categorical aspects of mod $p$ principal series representations and their centrality in modern approaches to the modular representation theory of reductive groups over local fields, finite rings, and their metaplectic covers.